CHAPTER 7
INTRODUCTION TO CONFIDENCE INTERVALS
We now begin our study into inferential statistics.
We are going to use statistics to infer what the data is telling
us.
In order to do so, we will use the normal approximation to the binomial, normal distribution, t-distribution, and
2
χ
distribution.
We have only seen two (2) of these. We will need to learn about the other distributions, first.
MORE DISTRIBUTIONS
T-Distribution
The t-distribution is similar to the standard normal distribution.
It will be needed in this chapter.
It shape
resembles the normal distribution.
This includes having equal mean, median and mode; being symmetric about the mean;
bell-shaped; and never touching the x-axis.
In fact, it is hard to tell the difference between the two when looking at the
histograms or distribution curves.
The t-distribution does not share the property of variance=1.
The t-distribution uses
something called degrees of freedom.
The
degrees of freedom (d.f.)
is equal to the sample size minus 1 (
1
−
n
).
(One
way to look at it:
If I needed to calculate that same mean again using the same sample size, and if I know all but 1 of the
values, I can figure out what the last value must equal to give that same mean.
The degrees of freedom are what I have to
play around with to get that mean again.
I cannot play with the last value.)
The degrees of freedom change the t-
distribution’s curve.
As the degrees of freedom get large, the t-distribution gets closer and closer to the standard normal,
z
.
The t-distribution is Table F.
It is on the pull-out reference card and pg. 784 (A-33) of the text.
Ex.)
Find
05
.
0
t
with 15 degrees of freedom.
Go to Table F.
The subscript 0.05 is the one tail
α
.
Underneath the one tail column that has 0.05, scroll
your hand down until you get to 15 on the far left (under d.f.).
1.753
**It is also worth mentioning the last line of Table F.
Underneath the degrees of freedom it says
∞
)
(
z
.
This is
because under a large (or infinite) number of degrees of freedom, the values you obtain from the t-table are the same as
that from the z-table.
Hence, the footnotes at the bottom of the table specifying what the values were rounded to for the z-
values.
Chi-Squared Distribution (denoted
2
χ
& pronounced ki-squared)
The chi-squared distribution is a positive (right) skewed continuous distribution.
Unlike the normal distribution, it
cannot take negative values.
It does not have symmetric properties.
Chi-square distribution uses degrees of freedom, as
did the T-distribution.
The degrees of freedom change the curve, also.
The degrees of freedom equal the sample size
minus 1
(n-1).
The chi-squared distribution is Table G pg. 772 (A-34).
Ex.)
Find
2
025
.
0
χ
with n=10.
Degrees of freedom = 10 -1 = 9.
Match up the
α
=0.025 with d.f.=9.
19.023
ESTIMATORS
In inferential statistics, we are going to use statistics as
estimators
of the population parameters.
For example, we
use the sample mean as an estimate of the population mean.
We are using a single value to estimate the parameter,
population mean.
That is called a
point estimator.

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